New modularity bounds for graphs $G(n,r,s)$ and $G_p(n,r,s)$

We analyze the behavior of the modularity of $G(n,r,s)$ graphs in the case of $r=o(\sqrt{{n}})$ and $n\to\infty$ and also that of $G_p(n,r,s)$ graphs for fixed $r$ and $s$ as $n\to\infty$. For $G(n,r,s)$ graphs with $r\ge cs^2$, we obtain substantial improvements of previously known upper bounds. Upper and lower bounds previously obtained for $G(n,r,s)$ graphs are extended to the family of $G_p(n,r,s)$ graphs with $p=p(n)=\omega\bigl(n^{-\frac{r-s-1}{2}}\bigr)$ and fixed $r$ and $s$.